Spherically Balanced Hilbert Spaces of Formal Power Series in Several Variables-II

Abstract

We continue the study of spherically balanced Hilbert spaces initiated in the first part of this paper. Recall that the complex Hilbert space $H^2(\beta )$ of formal power series in the variables $z_1, \ldots , z_m$ is spherically balanced if and only if there exist a Reinhardt measure $\mu$ supported on the unit sphere $\partial {\mathbb {B}}$ and a Hilbert space $H^2(\gamma )$ of formal power series in the variable $t$ such that

$$\begin{aligned} \Vert f\Vert ^2_{H^2(\beta )} = \int _{\partial {\mathbb {B}}}\Vert {f_z}\Vert ^2_{H^2(\gamma )}~d\mu (z)~(f \in H^2(\beta )), \end{aligned}$$

where $f_z(t)=f(t z)$ is a formal power series in the variable $t$. In the first half of this paper, we discuss operator theory in spherically balanced Hilbert spaces. The first main result in this part describes quasi-similarity orbit of multiplication tuple $M_z$ on a spherically balanced space $H^2(\beta ).$ We also observe that all spherical contractive multi-shifts on spherically balanced spaces admit the classical von Neumann’s inequality. In the second half, we introduce and study a class of Hilbert spaces, to be referred to as ${\mathcal {G}}$-balanced Hilbert spaces, where ${\mathcal {G}}={\mathcal {U}}(r_1) \times {\mathcal {U}}(r_2) \times \cdots \times {\mathcal {U}}(r_k)$ is a subgroup of ${\mathcal {U}}(m)$ with $r_1 + \cdots + r_k=m.$ In the case in which ${\mathcal {G}}={\mathcal {U}}(m),$ ${\mathcal {G}}$-balanced spaces are precisely spherically balanced Hilbert spaces.